3.366 \(\int \text{sech}^5(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\)
Optimal. Leaf size=126 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{8 f \sqrt{a-b}}+\frac{\tanh (e+f x) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 f}+\frac{3 a \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 f} \]
[Out]
(3*a^2*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*Sqrt[a - b]*f) + (3*a*Sech[e + f*x]
*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(8*f) + (Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*
x])/(4*f)
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Rubi [A] time = 0.120855, antiderivative size = 126, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used =
{3190, 378, 377, 203} \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{8 f \sqrt{a-b}}+\frac{\tanh (e+f x) \text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 f}+\frac{3 a \tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{8 f} \]
Antiderivative was successfully verified.
[In]
Int[Sech[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
(3*a^2*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*Sqrt[a - b]*f) + (3*a*Sech[e + f*x]
*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(8*f) + (Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*
x])/(4*f)
Rule 3190
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rule 378
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \text{sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{\left (1+x^2\right )^3} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 f}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{\left (1+x^2\right )^2} \, dx,x,\sinh (e+f x)\right )}{4 f}\\ &=\frac{3 a \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 f}+\frac{\text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 f}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{8 f}\\ &=\frac{3 a \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 f}+\frac{\text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 f}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{8 f}\\ &=\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{8 \sqrt{a-b} f}+\frac{3 a \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 f}+\frac{\text{sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 f}\\ \end{align*}
Mathematica [C] time = 0.108615, size = 66, normalized size = 0.52 \[ \frac{a^2 \sinh (e+f x) \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};-\frac{(a-b) \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}\right )}{f \sqrt{a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
(a^2*Hypergeometric2F1[1/2, 3, 3/2, -(((a - b)*Sinh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2))]*Sinh[e + f*x])/(f*Sq
rt[a + b*Sinh[e + f*x]^2])
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Maple [C] time = 0.107, size = 63, normalized size = 0.5 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{{b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+2\,ab \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+{a}^{2}}{ \left ( \cosh \left ( fx+e \right ) \right ) ^{6}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x)
[Out]
`int/indef0`((b^2*sinh(f*x+e)^4+2*a*b*sinh(f*x+e)^2+a^2)/cosh(f*x+e)^6/(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/
f
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{sech}\left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^5, x)
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Fricas [B] time = 3.80103, size = 7839, normalized size = 62.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
[-1/16*(3*(a^2*cosh(f*x + e)^8 + 8*a^2*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*sinh(f*x + e)^8 + 4*a^2*cosh(f*x +
e)^6 + 4*(7*a^2*cosh(f*x + e)^2 + a^2)*sinh(f*x + e)^6 + 6*a^2*cosh(f*x + e)^4 + 8*(7*a^2*cosh(f*x + e)^3 + 3*
a^2*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*a^2*cosh(f*x + e)^4 + 30*a^2*cosh(f*x + e)^2 + 3*a^2)*sinh(f*x + e)
^4 + 4*a^2*cosh(f*x + e)^2 + 8*(7*a^2*cosh(f*x + e)^5 + 10*a^2*cosh(f*x + e)^3 + 3*a^2*cosh(f*x + e))*sinh(f*x
+ e)^3 + 4*(7*a^2*cosh(f*x + e)^6 + 15*a^2*cosh(f*x + e)^4 + 9*a^2*cosh(f*x + e)^2 + a^2)*sinh(f*x + e)^2 + a
^2 + 8*(a^2*cosh(f*x + e)^7 + 3*a^2*cosh(f*x + e)^5 + 3*a^2*cosh(f*x + e)^3 + a^2*cosh(f*x + e))*sinh(f*x + e)
)*sqrt(-a + b)*log(((a - 2*b)*cosh(f*x + e)^4 + 4*(a - 2*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a - 2*b)*sinh(f*x
+ e)^4 - 2*(3*a - 2*b)*cosh(f*x + e)^2 + 2*(3*(a - 2*b)*cosh(f*x + e)^2 - 3*a + 2*b)*sinh(f*x + e)^2 - 2*sqrt
(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(-a + b)*sqrt((b*cosh(f*x + e)
^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a
- 2*b)*cosh(f*x + e)^3 - (3*a - 2*b)*cosh(f*x + e))*sinh(f*x + e) + a - 2*b)/(cosh(f*x + e)^4 + 4*cosh(f*x +
e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cosh(f*x + e)^2 + 4*(cosh
(f*x + e)^3 + cosh(f*x + e))*sinh(f*x + e) + 1)) - 2*sqrt(2)*((3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^6 + 6*(3*a^2
- a*b - 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*a^2 - a*b - 2*b^2)*sinh(f*x + e)^6 + (11*a^2 - 17*a*b + 6*b
^2)*cosh(f*x + e)^4 + (15*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^2 + 11*a^2 - 17*a*b + 6*b^2)*sinh(f*x + e)^4 + 4
*(5*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^3 + (11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - (11*a^2
- 17*a*b + 6*b^2)*cosh(f*x + e)^2 + (15*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^4 + 6*(11*a^2 - 17*a*b + 6*b^2)*c
osh(f*x + e)^2 - 11*a^2 + 17*a*b - 6*b^2)*sinh(f*x + e)^2 - 3*a^2 + a*b + 2*b^2 + 2*(3*(3*a^2 - a*b - 2*b^2)*c
osh(f*x + e)^5 + 2*(11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e)^3 - (11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e))*sinh(f
*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x +
e) + sinh(f*x + e)^2)))/((a - b)*f*cosh(f*x + e)^8 + 8*(a - b)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a - b)*f*sin
h(f*x + e)^8 + 4*(a - b)*f*cosh(f*x + e)^6 + 4*(7*(a - b)*f*cosh(f*x + e)^2 + (a - b)*f)*sinh(f*x + e)^6 + 6*(
a - b)*f*cosh(f*x + e)^4 + 8*(7*(a - b)*f*cosh(f*x + e)^3 + 3*(a - b)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35
*(a - b)*f*cosh(f*x + e)^4 + 30*(a - b)*f*cosh(f*x + e)^2 + 3*(a - b)*f)*sinh(f*x + e)^4 + 4*(a - b)*f*cosh(f*
x + e)^2 + 8*(7*(a - b)*f*cosh(f*x + e)^5 + 10*(a - b)*f*cosh(f*x + e)^3 + 3*(a - b)*f*cosh(f*x + e))*sinh(f*x
+ e)^3 + 4*(7*(a - b)*f*cosh(f*x + e)^6 + 15*(a - b)*f*cosh(f*x + e)^4 + 9*(a - b)*f*cosh(f*x + e)^2 + (a - b
)*f)*sinh(f*x + e)^2 + (a - b)*f + 8*((a - b)*f*cosh(f*x + e)^7 + 3*(a - b)*f*cosh(f*x + e)^5 + 3*(a - b)*f*co
sh(f*x + e)^3 + (a - b)*f*cosh(f*x + e))*sinh(f*x + e)), 1/8*(3*(a^2*cosh(f*x + e)^8 + 8*a^2*cosh(f*x + e)*sin
h(f*x + e)^7 + a^2*sinh(f*x + e)^8 + 4*a^2*cosh(f*x + e)^6 + 4*(7*a^2*cosh(f*x + e)^2 + a^2)*sinh(f*x + e)^6 +
6*a^2*cosh(f*x + e)^4 + 8*(7*a^2*cosh(f*x + e)^3 + 3*a^2*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*a^2*cosh(f*x
+ e)^4 + 30*a^2*cosh(f*x + e)^2 + 3*a^2)*sinh(f*x + e)^4 + 4*a^2*cosh(f*x + e)^2 + 8*(7*a^2*cosh(f*x + e)^5 +
10*a^2*cosh(f*x + e)^3 + 3*a^2*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*a^2*cosh(f*x + e)^6 + 15*a^2*cosh(f*x + e
)^4 + 9*a^2*cosh(f*x + e)^2 + a^2)*sinh(f*x + e)^2 + a^2 + 8*(a^2*cosh(f*x + e)^7 + 3*a^2*cosh(f*x + e)^5 + 3*
a^2*cosh(f*x + e)^3 + a^2*cosh(f*x + e))*sinh(f*x + e))*sqrt(a - b)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f
*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(a - b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b
)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*
sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 2*a - b)*sinh(f*x
+ e)^2 + 4*(b*cosh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*x + e) + b)) + sqrt(2)*((3*a^2 - a*b - 2*b^2)
*cosh(f*x + e)^6 + 6*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*a^2 - a*b - 2*b^2)*sinh(f*x + e)
^6 + (11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e)^4 + (15*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^2 + 11*a^2 - 17*a*b +
6*b^2)*sinh(f*x + e)^4 + 4*(5*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^3 + (11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e)
)*sinh(f*x + e)^3 - (11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e)^2 + (15*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^4 + 6*
(11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e)^2 - 11*a^2 + 17*a*b - 6*b^2)*sinh(f*x + e)^2 - 3*a^2 + a*b + 2*b^2 + 2
*(3*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)^5 + 2*(11*a^2 - 17*a*b + 6*b^2)*cosh(f*x + e)^3 - (11*a^2 - 17*a*b + 6
*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 -
2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/((a - b)*f*cosh(f*x + e)^8 + 8*(a - b)*f*cosh(f*x + e)*sinh
(f*x + e)^7 + (a - b)*f*sinh(f*x + e)^8 + 4*(a - b)*f*cosh(f*x + e)^6 + 4*(7*(a - b)*f*cosh(f*x + e)^2 + (a -
b)*f)*sinh(f*x + e)^6 + 6*(a - b)*f*cosh(f*x + e)^4 + 8*(7*(a - b)*f*cosh(f*x + e)^3 + 3*(a - b)*f*cosh(f*x +
e))*sinh(f*x + e)^5 + 2*(35*(a - b)*f*cosh(f*x + e)^4 + 30*(a - b)*f*cosh(f*x + e)^2 + 3*(a - b)*f)*sinh(f*x +
e)^4 + 4*(a - b)*f*cosh(f*x + e)^2 + 8*(7*(a - b)*f*cosh(f*x + e)^5 + 10*(a - b)*f*cosh(f*x + e)^3 + 3*(a - b
)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a - b)*f*cosh(f*x + e)^6 + 15*(a - b)*f*cosh(f*x + e)^4 + 9*(a - b)
*f*cosh(f*x + e)^2 + (a - b)*f)*sinh(f*x + e)^2 + (a - b)*f + 8*((a - b)*f*cosh(f*x + e)^7 + 3*(a - b)*f*cosh(
f*x + e)^5 + 3*(a - b)*f*cosh(f*x + e)^3 + (a - b)*f*cosh(f*x + e))*sinh(f*x + e))]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)**5*(a+b*sinh(f*x+e)**2)**(3/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{sech}\left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^5, x)